The number of hours five students spend on homework per week are 4, 6, 8, 10, and 12. If each number is decreased by 2, what is the variance of the new data set?
Show the answer and explanation
Correct answer
A. 8
Principle or equation
Subtracting a constant from each data value does not change the variance, because variance measures spread and is invariant under shifts.
Why this answer is correct
Original data: 4,6,8,10,12. Mean = 8. Deviations: -4,-2,0,2,4. Squared deviations: 16,4,0,4,16. Sum = 40. Population variance = 40/5 = 8. After subtracting 2, data: 2,4,6,8,10. Mean = 6. Deviations: -4,-2,0,2,4. Same squared deviations, sum = 40, variance = 8. Alternatively, variance unchanged by constant shift.
Example
For data 1,3,5, variance = (4+0+4)/3 = 8/3. Subtract 1: 0,2,4, variance still 8/3.
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